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''of a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817801.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817802.png" />''
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A number corresponding to each one-dimensional subspace of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817803.png" /> by the formula
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817804.png" /></td> </tr></table>
+
''of a Riemannian manifold  $  M $
 +
at a point  $  p \in M $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817805.png" /> is the [[Ricci tensor|Ricci tensor]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817806.png" /> is a vector generating the one-dimensional subspace and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817807.png" /> is the [[Metric tensor|metric tensor]] of the [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817808.png" />. The Ricci curvature can be expressed in terms of the sectional curvatures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r0817809.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178010.png" /> be the [[Sectional curvature|sectional curvature]] at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178011.png" /> in the direction of the surface element defined by the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178013.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178014.png" /> be normalized vectors orthogonal to each other and to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178015.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178016.png" /> be the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178017.png" />; then
+
A number corresponding to each one-dimensional subspace of the tangent space  $  M _ {p} $
 +
by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178018.png" /></td> </tr></table>
+
$$
 +
r ( v)  = \
  
For manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178019.png" /> of dimension greater than two the following proposition is valid: If the Ricci curvature at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178020.png" /> has one and the same value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178021.png" /> in all directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178022.png" />, then the Ricci curvature has one and the same value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178023.png" /> at all points of the manifold. Manifolds of constant Ricci curvature are called Einstein spaces. The Ricci tensor of an Einstein space is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178025.png" /> is the Ricci curvature. For an Einstein space the following equality holds:
+
\frac{( c R ) ( v , v ) }{g ( v , v ) }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178026.png" /></td> </tr></table>
+
where  $  c R $
 +
is the [[Ricci tensor|Ricci tensor]],  $  v $
 +
is a vector generating the one-dimensional subspace and  $  g $
 +
is the [[Metric tensor|metric tensor]] of the [[Riemannian manifold|Riemannian manifold]]  $  M $.  
 +
The Ricci curvature can be expressed in terms of the sectional curvatures of  $  M $.  
 +
Let  $  K _ {p} ( \alpha , \beta ) $
 +
be the [[Sectional curvature|sectional curvature]] at the point  $  p \in M $
 +
in the direction of the surface element defined by the vectors  $  \alpha $
 +
and  $  \beta $,
 +
let  $  l _ {1} \dots l _ {n-} 1 $
 +
be normalized vectors orthogonal to each other and to the vector  $  v $,
 +
and let  $  n $
 +
be the dimension of  $  M $;
 +
then
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178028.png" /> are the covariant and contravariant components of the Ricci tensor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178029.png" /> is the dimension of the space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178030.png" /> is the scalar curvature of the space.
+
$$
 +
r ( v)  = \
 +
\sum _ { i= } 1 ^ { n- }  1 K _ {p} ( v , l _ {i} ) .
 +
$$
 +
 
 +
For manifolds  $  M $
 +
of dimension greater than two the following proposition is valid: If the Ricci curvature at a point  $  p \in M $
 +
has one and the same value  $  r $
 +
in all directions  $  v $,
 +
then the Ricci curvature has one and the same value  $  r $
 +
at all points of the manifold. Manifolds of constant Ricci curvature are called Einstein spaces. The Ricci tensor of an Einstein space is of the form  $  c R = r g $,
 +
where  $  r $
 +
is the Ricci curvature. For an Einstein space the following equality holds:
 +
 
 +
$$
 +
n R _ {ij} R  ^ {ij} - s  ^ {2}  = 0 ,
 +
$$
 +
 
 +
where  $  R _ {ij} $,
 +
$  R  ^ {ij} $
 +
are the covariant and contravariant components of the Ricci tensor, $  n $
 +
is the dimension of the space and $  s $
 +
is the scalar curvature of the space.
  
 
The Ricci curvature can be defined by similar formulas also on pseudo-Riemannian manifolds; in this case the vector is assumed to be anisotropic.
 
The Ricci curvature can be defined by similar formulas also on pseudo-Riemannian manifolds; in this case the vector is assumed to be anisotropic.
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From the Ricci curvature the Ricci tensor can be recovered uniquely:
 
From the Ricci curvature the Ricci tensor can be recovered uniquely:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178031.png" /></td> </tr></table>
+
$$
 +
( c R ) ( u , v ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081780/r08178032.png" /></td> </tr></table>
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$$
 +
= \
 +
 
 +
\frac{1}{2}
 +
[ r ( u + v ) g ( u + v , u + v )
 +
- r ( u) g ( u , u ) - r ( v) g ( v , v ) ] .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Z. Petrov,  "Einstein spaces" , Pergamon  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Z. Petrov,  "Einstein spaces" , Pergamon  (1969)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.L. Besse,  "Einstein manifolds" , Springer  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.J. Hicks,  "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.L. Besse,  "Einstein manifolds" , Springer  (1987)</TD></TR></table>

Revision as of 08:11, 6 June 2020


of a Riemannian manifold $ M $ at a point $ p \in M $

A number corresponding to each one-dimensional subspace of the tangent space $ M _ {p} $ by the formula

$$ r ( v) = \ \frac{( c R ) ( v , v ) }{g ( v , v ) } , $$

where $ c R $ is the Ricci tensor, $ v $ is a vector generating the one-dimensional subspace and $ g $ is the metric tensor of the Riemannian manifold $ M $. The Ricci curvature can be expressed in terms of the sectional curvatures of $ M $. Let $ K _ {p} ( \alpha , \beta ) $ be the sectional curvature at the point $ p \in M $ in the direction of the surface element defined by the vectors $ \alpha $ and $ \beta $, let $ l _ {1} \dots l _ {n-} 1 $ be normalized vectors orthogonal to each other and to the vector $ v $, and let $ n $ be the dimension of $ M $; then

$$ r ( v) = \ \sum _ { i= } 1 ^ { n- } 1 K _ {p} ( v , l _ {i} ) . $$

For manifolds $ M $ of dimension greater than two the following proposition is valid: If the Ricci curvature at a point $ p \in M $ has one and the same value $ r $ in all directions $ v $, then the Ricci curvature has one and the same value $ r $ at all points of the manifold. Manifolds of constant Ricci curvature are called Einstein spaces. The Ricci tensor of an Einstein space is of the form $ c R = r g $, where $ r $ is the Ricci curvature. For an Einstein space the following equality holds:

$$ n R _ {ij} R ^ {ij} - s ^ {2} = 0 , $$

where $ R _ {ij} $, $ R ^ {ij} $ are the covariant and contravariant components of the Ricci tensor, $ n $ is the dimension of the space and $ s $ is the scalar curvature of the space.

The Ricci curvature can be defined by similar formulas also on pseudo-Riemannian manifolds; in this case the vector is assumed to be anisotropic.

From the Ricci curvature the Ricci tensor can be recovered uniquely:

$$ ( c R ) ( u , v ) = $$

$$ = \ \frac{1}{2} [ r ( u + v ) g ( u + v , u + v ) - r ( u) g ( u , u ) - r ( v) g ( v , v ) ] . $$

References

[1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2] A.Z. Petrov, "Einstein spaces" , Pergamon (1969) (Translated from Russian)

Comments

References

[a1] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a2] A.L. Besse, "Einstein manifolds" , Springer (1987)
How to Cite This Entry:
Ricci curvature. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_curvature&oldid=48536
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article